Newcastle University e-prints Date deposited: 12th October 2011 Version of file: Author final Peer Review Status: Peer reviewed Citation for item: Chakraborty N, Hartung G, Katragadda M, Kaminski CF. Comparison of 2D and 3D density-weighted displacement speed statistics and implications for laser based measurements of flame displacement speed using direct numerical simulation data. Combustion and Flame 2011, 158(7), 1372-1390. Further information on publisher website: http://www.elsevier.com Publisher’s copyright statement: The definitive version of this article, published by Elsevier, 2011, is available at: http://dx.doi.org/10.1016/j.combustflame.2010.11.014 Always use the definitive version when citing. Use Policy: The full-text may be used and/or reproduced and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not for profit purposes provided that: • A full bibliographic reference is made to the original source • A link is made to the metadata record in Newcastle E-prints • The full text is not changed in any way. The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Robinson Library, University of Newcastle upon Tyne, Newcastle upon Tyne. NE1 7RU. Tel. 0191 222 6000 Comparison of 2D and 3D density-weighted displacement speed statistics and implications for laser based measurements of flame displacement speed using Direct Numerical Simulation data N. Chakrabortya*, G. Hartung b, M. Katragaddac, and, C. F. Kaminskib,d a University of Liverpool, Engineering Department, Brownlow Hill,L69 3GH, UK Email: [email protected] b University of Cambridge, Department of Chemical Engineering, Pembroke Street, Cambridge, CB2 3RA, UK Email: [email protected] c University of Liverpool, Engineering Department, Brownlow Hill,L69 3GH, UK Email: [email protected] d SAOT School of Advanced Optical Technologies, Friedrich Alexander University Erlangen Nuremberg, 91058 Erlangen, Germany Email: [email protected] Running title: Comparison of 2D and 3D displacement speed statistics for premixed flames * Corresponding author ABSTRACT In a recent study a light sheet imaging approach has been proposed (Hartung et al., J. App. Phys. B, 96 2D (2009) 843-862) which permits measurement of a quantity S d , which is the two-dimensional projection of the actual density-weighted displacement speed S d for turbulent premixed flames. Here 2D the statistics of S d and S d are compared using a Direct Numerical Simulation database of statistically 2D planar turbulent premixed flames. It is found that the probability density functions (pdfs) of S d approximate the pdfs of Sd satisfactorily for small values of root-mean-square turbulent velocity 2D fluctuation u , though the S d pdfs are wider than the Sd pdfs. Although the agreement between the 2D pdfs and the standard deviations of S d and Sd deteriorate with increasing u , the mean values of 2D S d correspond closely with the mean values of S d for all cases considered here. The pdfs of two- 2D dimensional curvature m and the two-dimensional tangential-diffusion component of density-weighted 2D displacement speed St are found to be narrower than their three-dimensional counterparts (i.e. m 2D and St respectively). It has been found that the pdfs, mean and standard deviation of / 2 m and 2D / 2 St faithfully capture the pdfs, mean and standard deviation of the corresponding three- 2D dimensional counterparts, m and St respectively. The combination of wider S d pdfs in comparison 2D 2D 2D 2D to S d pdfs, and narrower St pdfs in comparison to St pdfs, leads to wider (Sr Sn ) Sd St pdfs than the pdfs of combined reaction and normal-diffusion components of density-weighted 2D displacement speed (S r Sn ) . This is reflected in the higher value of standard deviation of (Sr Sn ) , 2D than that of its three-dimensional counterpart (S r S n ) . However, the mean values of (S r Sn ) remain close to the mean values of (S r S n ) . The loss of perfect correlation between two and three- 2D 2D dimensional quantities leads to important qualitative differences between the (Sr Sn ) m and 2D 2D (Sr Sn )m , and between the Sd m and Sd m correlations. For unity Lewis number flames, the 2D Sd m correlation remains strongly negative, whereas a weak correlation is observed between Sd and 2D m . The study demonstrates the strengths and limitations of the predictive capabilities of the planar imaging techniques in the context of the measurement of density-weighted displacement speed, which are important for detailed model development or validation based on experimental data. Keywords: Turbulent premixed flame, Density-weighted displacement speed, curvature, Direct Numerical Simulations 2 NOMENCLATURE Arabic CP Specific heat at constant pressure CV Specific heat at constant volume c Reaction progress variable c Progress variable value defining flame surface D Progress variable diffusivity D0 Progress variable diffusivity in the unburned gas Da Damköhler number hmax Greater of principal curvatures by magnitude hmin Smaller of principal curvatures by magnitude k global Global turbulent kinetic energy evaluated over the whole domain k global,0 Global turbulent kinetic energy evaluated over the whole domain at initial time Ka Karlovitz number Le Lewis number l Integral length scale Ma Mach number N Actual flame normal vector in three-dimensions N 2D Apparent flame normal vector in two-dimensions 3 th N i i component of flame normal Pr Prandtl number Ret Turbulent Reynolds number Sc Schmidt number S d Actual displacement speed in three-dimensions S d Actual density-weighted displacement speed in three-dimensions 2D S d Apparent displacement speed in two-dimensions 2D S d Apparent density-weighted displacement speed in two-dimensions S L Laminar burning velocity S n Actual normal diffusion component of density-weighted displacement speed in three-dimensions St Actual tangential diffusion component of density-weighted displacement speed in three-dimensions 2D St Apparent tangential diffusion component of density-weighted displacement speed in two-dimensions S r Actual reaction component of density-weighted displacement speed in three-dimensions 2D (S r S n ) Apparent combined reaction and normal diffusion component of density- weighted displacement speed in two-dimensions 4 sh Shape factor tsim Simulation time Tˆ Dimensional temperature Tac Activation temperature Tad Adiabatic flame temperature T0 Reactant temperature th ui i component of fluid velocity u Root mean square turbulent velocity fluctuation v Kolmogorov velocity scale w Chemical reaction rate xi i th Cartesian co-ordinate Greek Angle determining local flame normal orientation Angle determining local flame normal orientation Z Zel’dovich number Ratio of specific heats (=CP /)CV th Thermal laminar flame thickness Kolmogorov length scale 5 m Actual flame curvature in three-dimensions 2D m Apparent flame curvature in two-dimensions 1 , 2 Principal curvatures Thermal conductivity Dynamic viscosity SD Mean value of S d / S L 2D 2D SD Mean value of S d / S L SRN Mean value of (Sr Sn ) / S L 2D 2D SRN Mean value of (S r Sn ) / S L Density F Density at the flame front 0 Unburned gas density SD Standard deviation of S d / S L 2D 2D SD Standard deviation of S d / S L SRN Standard deviation of (S r S n ) / S L 2D 2D SRN Standard-deviations of (S r S n ) / S L Flame Surface Density based on fine-grained description gen Generalised Flame Surface Density Heat release parameter 6 Acronyms DNS Direct Numerical Simulation PLIF Planar Laser Induced Fluorescence 2D Two-dimensional/ Two dimensions 3D Three-dimensional/ Three dimensions 7 1. INTRODUCTION The displacement speed S d is a quantity of key importance in turbulent premixed combustion [1], which represents the speed with which the flame front moves normal to itself with respect to an initially coincident material surface. Although S d is a quantity on which turbulent combustion models based on level-set [1] and Flame Surface Density (FSD) methodologies [2,3] critically depend, experimental data on the statistical behaviour of S d in turbulent flames are rarely presented in the literature [4,5]. For turbulent premixed flames S d is an inherently three-dimensional quantity and its determination requires the resolution of local flow velocity fields and complete knowledge of the evolving flame geometry. This requires a time-resolved measurement of the three-dimensional flame topology, as well as the full convective flow field. Despite all advances in laser imaging technology for turbulent combustion, attempts to measure three-dimensional displacement speed S d remain futile. Even if the future brings about the theoretical possibility of measuring S d , the associated uncertainties will, in all likelihood, limit the usefulness of the measurements. In a recent paper, Hartung et al. [5] presented an alternative methodology of measuring a quantity related to the density-weighted displacement speed S d F Sd / 0 based on previously developed techniques for the time-resolved planar imaging of the flame front contour [6-12] via Laser Induced Fluorescence (LIF) of OH and simultaneously performed 2D stereoscopic Particle Image Velocimetry (PIV). This yields a quantity S d which can be thought of as a 2D projection of S d onto the plane defined by the laser sheet. The quantity S d can be interpreted as a two- dimensional, density-weighted displacement speed, which can potentially be used for calibrating and validating turbulent combustion models. It was indicated in Ref. [5] that for flames with symmetry (e.g. 2D jet flames with statistical symmetry around the jet axis) the statistics of S d may represent the true 2D statistics of S d under certain conditions. As S d can be extracted from experimental data with relative 8 2D ease and high accuracy, it is important to assess the differences between S d and its actual three- dimensional counterpart.